258 



Mr. G. H. Darwin. 



[Mar. 18. 



The suffix m 2 to the symbols i and N there indicates that the equa- 

 tions (80) only refer to the action of the moon, and as here we only 

 have a single satellite, they are the complete equations. N is equal to 

 v/n , so that n disappears from the first and second of (80) ; also 

 fi=l/sn Q Q i, and thns n disappears from the third equation. P = cos?', 

 Q = sin?', and, since we are treating i the obliquity as small, P=l, 

 Q — i\ also \=Q/n; the e of that paper is identical with the /of the 

 present one ; lastly £ is equal to Q %Q - ^, and since with our present 

 units 5=1, therefore /ti dg/dt=dQ~*ln dt=dxln dt. 



With regard to the transformation of the first of (80) into (4) of 

 the present paper, I remark that treating i as small jPQ—^Q 

 =±i(l — 2Qjn), and introducing this transformation into the first of 

 (80), equation (4) is obtained, except that i occurs in place of (i+j). 

 Now in the paper on the " Precession of a Viscous Spheroid " the in- 

 clination of the orbit of the satellite to the plane of reference was 

 treated as zero, and hence j was zero ; but I have proved in a paper 

 " On the Secular Changes in the Elements of the Orbit of a Satellite 

 revolving about a tidally distorted Planet" (read before the Royal 

 Society on December 18th, 1879, but as yet unpublished) that when 

 we take into account the inclination of the orbit of the satellite, the P 

 and Q on the right-hand sides of eq. (80) of " Precession " must be 

 taken as the cosine and sine of i +j instead of i. Equations (5) and 

 (6) are proved in § 10, Part II, and § 25, Part V of the unpublished 

 paper, and the reader is requested to take them as established. 



The integrals of this system of equations will give the secular 

 changes in the motion of the system under the influence of the 

 fractional tides. The object of the present paper is to find an analy- 

 tical expression for the solution, and to interpret that solution geome- 

 trically. 



From equations (2) and (4) we have 



^+4i=iv sin v r~ <<+/> ° y 



at dt g L v n /_ 



But from (3) and (5) x dj/dt+j dxjdt is equal to the same expres- 

 sion; hence 



. dn , di dj , . dx 

 dt dt dt J dt 



The integral of this equation is in—jx. 



-.=- (7). 



J n 



Equation (7) may also be obtained by the principle of conservation 

 of moment of momentum. The motion is referred to the invariable 

 plane of the system, and however the planet and satellite may interact 

 on one another, the resultant m. of m. must remain constant in 



